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Theorem mulclpr 7570
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
Assertion
Ref Expression
mulclpr ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)

Proof of Theorem mulclpr
Dummy variables 𝑞 𝑟 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-imp 7467 . . . 4 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}⟩)
21genpelxp 7509 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 mulclnq 7374 . . . 4 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
41, 3genpml 7515 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)))
51, 3genpmu 7516 . . 3 ((𝐴P𝐵P) → ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))
62, 4, 5jca32 310 . 2 ((𝐴P𝐵P) → ((𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
7 ltmnqg 7399 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
8 mulcomnqg 7381 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulnqprl 7566 . . . . 5 ((((𝐴P𝑢 ∈ (1st𝐴)) ∧ (𝐵P𝑡 ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑢 ·Q 𝑡) → 𝑥 ∈ (1st ‘(𝐴 ·P 𝐵))))
101, 3, 7, 8, 9genprndl 7519 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))))
11 mulnqpru 7567 . . . . 5 ((((𝐴P𝑢 ∈ (2nd𝐴)) ∧ (𝐵P𝑡 ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑢 ·Q 𝑡) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 ·P 𝐵))))
121, 3, 7, 8, 11genprndu 7520 . . . 4 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1310, 12jca 306 . . 3 ((𝐴P𝐵P) → (∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))))
141, 3, 7, 8genpdisj 7521 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))
15 mullocpr 7569 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1613, 14, 153jca 1177 . 2 ((𝐴P𝐵P) → ((∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵))))))
17 elnp1st2nd 7474 . 2 ((𝐴 ·P 𝐵) ∈ P ↔ (((𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))) ∧ ((∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))))
186, 16, 17sylanbrc 417 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978  wcel 2148  wral 2455  wrex 2456  𝒫 cpw 3575   class class class wbr 4003   × cxp 4624  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   ·Q cmq 7281   <Q cltq 7283  Pcnp 7289   ·P cmp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-imp 7467
This theorem is referenced by:  mulnqprlemfl  7573  mulnqprlemfu  7574  mulnqpr  7575  mulassprg  7579  distrlem1prl  7580  distrlem1pru  7581  distrlem4prl  7582  distrlem4pru  7583  distrlem5prl  7584  distrlem5pru  7585  distrprg  7586  1idpr  7590  recexprlemex  7635  ltmprr  7640  mulcmpblnrlemg  7738  mulcmpblnr  7739  mulclsr  7752  mulcomsrg  7755  mulasssrg  7756  distrsrg  7757  m1m1sr  7759  1idsr  7766  00sr  7767  recexgt0sr  7771  mulgt0sr  7776  mulextsr1lem  7778  mulextsr1  7779  recidpirq  7856
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